Question

Find the second order derivatives of the following function e^x sin 5x  ?

Answer 1

We have,

\[y = e^x \sin\left( 5x \right)\]

\[\text { Differentiating w . r . t . x, we get }\]

\[\frac{d y}{d x} = e^x \sin 5x + e^x \cos 5x \times 5\]

\[\text { Differentiating again w . r . t . x, we get }\]

\[\frac{d^2 y}{d x^2} = e^x \sin 5x + e^x \cos 5x \times 5 + 5 e^x ( - \sin5x \times 5) + 5 e^x \cos 5x \]

\[ = - 24 e^x \sin 5x + 10 e^x \cos 5x\]

\[ = 2 e^x \left( 5 \cos 5x - 12 \sin 5x \right)\]